8.02 Spring 2009Experiment 9: Undriven RLC CircuitsINTRODUCTIONUndriven Circuits: Thinking about OscillationsFigure 2 Undriven RLC circuit. (a) For t<0 the switch S is open and although the capacitor is charged (Q = Q0) no current flows in the circuit. (b) A half period after closing the switch the capacitor again comes to a maximum charge, this time with the positive charge on the lower plate.2. AC/DC Electronics Lab Circuit BoardIn addition, in parallel with the capacitor you will connect a voltage probe (not pictured).GENERALIZED PROCEDUREIn this lab you will measure the behavior of an undriven series RLC circuit.Part 1 is repeated, except that the energy is reported instead of current and voltage.END OF PRE-LAB READINGIN-LAB ACTIVITIESEXPERIMENTAL SETUPMEASUREMENTSPart 1: Free Oscillations in an Undriven RLC CircuitPart 2: Energy Ringdown in an Undriven RLC CircuitMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2009 Experiment 9: Undriven RLC Circuits OBJECTIVES 1. To explore the time dependent behavior of Undriven RLC Circuits 2. To understand the idea of resonance PRE-LAB READING INTRODUCTION As most children know, if you get a push on a swing and just sit still on it, you will go back and forth, gradually slowing down to a stop. If, on the other hand, you move your body back and forth you can drive the swing, making it swing higher and higher. This only works if you move at the correct rate though – too fast or too slow and the swing will do nothing. This is an example of resonance in a mechanical system. In this lab we will explore its electrical analog – the RLC (resistor, inductor, capacitor) circuit – and better understand what happens when it is undriven. In the next lab we will consider what happens when it is driven above, below and at the resonant frequency. The Details: Oscillations In this lab you will be investigating current and voltages (EMFs) in RLC circuits. These oscillate as a function of time, either continuously (Fig. 1a) or in a decaying fashion (Fig. 1b). 0T 1T 2T-X0X0AmplitudeTime (in Periods) 0T 1T 2T 3T 4T 5T-X0X0AmplitudeTime (in Periods) (a) (b) Figure 1 Oscillating Functions. (a) A purely oscillating function ()0sinxx tωϕ=+ has fixed amplitude x0, angular frequency ω (period T = 2π/ω and frequency f = ω/2π), and phase φ (in this case φ = -0.2π). (b) The amplitude of a damped oscillating function decays exponentially (amplitude envelope indicated by dotted lines) E09-1Undriven Circuits: Thinking about Oscillations Consider the RLC circuit of fig. 2 below. The capacitor has an initial charge Q0 (it was charged by a battery no longer in the circuit), but it can’t go anywhere because the switch is open. When the switch is closed, the positive charge will flow off the top plate of the capacitor, through the resistor and inductor, and on to the bottom plate of the capacitor. This is the same behavior that we saw in RC circuits. In those circuits, however, the current flow stops as soon as all the positive charge has flowed to the negatively charged plate, leaving both plates with zero charge. The addition of an inductor, however, introduces inertia into the circuit, keeping the current flowing even when the capacitor is completely discharged, and forcing it to charge in the opposite polarity (Fig 2b). (a) (b) Figure 2 Undriven RLC circuit. (a) For t<0 the switch S is open and although the capacitor is charged (Q = Q0) no current flows in the circuit. (b) A half period after closing the switch the capacitor again comes to a maximum charge, this time with the positive charge on the lower plate. This oscillation of positive charge from the upper to lower plate of the capacitor is only one of the oscillations occurring in the circuit. For the two times pictured above (t=0 and t=0.5 T) the charge on the capacitor is a maximum and no current flows in the circuit. At intermediate times current is flowing, and, for example, at t = 0.25 T the current is a maximum and the charge on the capacitor is zero. Thus another oscillation in the circuit is between charge on the capacitor and current in the circuit. This corresponds to yet another oscillation in the circuit, that of energy between the capacitor and the inductor. When the capacitor is fully charged and the current is zero, the capacitor stores energy but the inductor doesn’t (2122; 0CLUQCU LI==2=). A quarter period later the current I is a maximum, charge Q = 0, and all the energy is in the inductor: 21220;CLUQC U LI===2. If there were no resistance in the circuit this swapping of energy between the capacitor and inductor would be perfect and the current (and voltage across the capacitor and EMF induced by the inductor) would oscillate as in Fig. 1a. A resistor, however, damps the circuit, removing energy by dissipating power through Joule heating (P=I2R), and eventually ringing the current down to zero, as in Fig. 1b. Note that only the resistor dissipates power. The capacitor and inductor both store energy during half the cycle and then completely release it during the other half. E09-2APPARATUS 1. Science Workshop 750 Interface In this lab we will again use the Science Workshop 750 interface as an AC function generator, whose voltage we can set and current we can measure. We will also use it to measure the voltage across the capacitor using a voltage probe. 2. AC/DC Electronics Lab Circuit Board E09-3 at left. We will also again use the circuit board, set up with a 100 μF capacitor in series with the coil (which serves both as the resistor and inductor in the circuit), as pictured Figure 3 Setup of the AC/DC Electronics Lab Circuit Board. In addition, in parallel with the capacitor you will connect a voltage probe (not pictured). GENERALIZED PROCEDURE In this lab you will measure the behavior of an undriven series RLC circuit. Part 1: Free Oscillations in an Undriven RLC Circuit The capacitor is charged with a DC battery which is then turned off, allowing the circuit to ring down. Part 2: Energy Ringdown in an Undriven RLC Circuit Part 1 is repeated, except that the energy is reported instead of current and voltage. END OF PRE-LAB READINGIN-LAB ACTIVITIES EXPERIMENTAL SETUP 1. Download the LabView file from the web and save the file to your desktop. Start LabView by double clicking on this file. 2. Set up the circuit pictured in Fig. 3 of the pre-lab reading
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